The spherical Bessel and Struve functions and operational methods

TL;DR

This paper reviews the theory of spherical Bessel and Struve functions using an operational umbral method that simplifies their evaluation and unifies various special functions.

Contribution

It introduces an operational approach that reduces complex functions to elementary Gaussian functions, unifying multiple special functions within a single formalism.

Findings

Provides a method for straightforward evaluation of integrals and derivatives of these functions.

Unifies spherical Bessel, Struve, Anger, and Weber functions under a common operational framework.

Discusses connections to multi-index Bessel functions.

Abstract

We review some aspects of the theory of spherical Bessel functions and Struve functions by means of an operational procedure essentially of umbral nature, capable of providing the straightforward evaluation of their definite integrals and of successive derivatives. The method we propose allows indeed the formal reduction of these family of functions to elementary ones of Gaussian type. We study the problem in general terms and present a formalism capable of providing a unifying point of view including Anger and Weber functions too. The link to the multi-index Bessel functions is also briefly discussed.